Functions with bounded Hessian-Schatten variation: density, variational and extremality properties

In this paper we analyze in detail a few questions related to the theory of functions with bounded \(p\)-Hessian-Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the \(p\)-Hessian-Scha...

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Bibliographic Details
Published in:arXiv.org 2023-02
Main Authors: Ambrosio, Luigi, Brena, Camillo, Conti, Sergio
Format: Article
Language:English
Subjects:
Continuity (mathematics)
Density
Inverse problem theory
Machine learning
Online Access:Get full text
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Summary:In this paper we analyze in detail a few questions related to the theory of functions with bounded \(p\)-Hessian-Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the \(p\)-Hessian-Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension \(d\), using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the \(p\)-Hessian-Schatten total variation are CPWL. Finally, we prove existence of minimizers of certain relevant functionals involving the \(p\)-Hessian-Schatten total variation in the critical dimension \(d=2\).
ISSN:2331-8422